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 Sibirsk. Mat. Zh., 2007, Volume 48, Number 1, Pages 205–213 (Mi smj17)

Powerful digraphs

S. V. Sudoplatov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type $p$ possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type $p$ and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (>1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (>1) countable models.

Keywords: powerful type, powerful digraph, infinite weight.

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English version:
Siberian Mathematical Journal, 2007, 48:1, 165–171

Bibliographic databases:

UDC: 510.67

Citation: S. V. Sudoplatov, “Powerful digraphs”, Sibirsk. Mat. Zh., 48:1 (2007), 205–213; Siberian Math. J., 48:1 (2007), 165–171

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. V. Sudoplatov, “Small Stable Generic Graphs with Infinite Weight. Bipartite Digraphs”, Siberian Adv. Math., 17:1 (2007), 37–48
2. S. V. Sudoplatov, “Complete Theories with Finitely Many Countable Models. II”, Algebra and Logic, 45:3 (2006), 180–200
3. S. V. Sudoplatov, “Small Stable Generic Graphs with Infinite Weight. Digraphs without Furcations”, Siberian Adv. Math., 18:2 (2008), 142–150
4. S. V. Sudoplatov, “On expansions and extensions of powerful digraphs”, Siberian Math. J., 50:3 (2009), 498–502
5. S. V. Sudoplatov, “Hypergraphs of prime models and distributions of countable models of small theories”, J. Math. Sci., 169:5 (2010), 680–695
6. I. V. Shulepov, S. V. Sudoplatov, “Algebras of distributions for isolating formulas of a complete theory”, Sib. elektron. matem. izv., 11 (2014), 380–407
7. S. V. Sudoplatov, “Combinations of structures”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 24 (2018), 82–101
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